Medical imaging is one of the most useful diagnostic tools available in modern medicine. Medical imaging allows medical personnel to non-intrusively look into a living body in order to detect and assess many types of injuries, diseases, conditions, etc. Medical imaging allows doctors and technicians to more easily and correctly make a diagnosis, decide on a treatment, prescribe medication, perform surgery or other treatments, etc.
There are medical imaging processes of many types and for many different purposes, situations, or uses. They commonly share the ability to create an image of a bodily region of a patient, and can do so non-invasively. Examples of some common medical imaging types are nuclear medical (NM) imaging such as positron emission tomography (PET) and single photon emission computed tomography (SPECT), electron-beam X-ray computed tomography (CT), magnetic resonance imaging (MRI), and ultrasound (US). Using these or other imaging types and associated machines, an image or series of images may be captured. Other devices may then be used to process the image in some fashion. Finally, a doctor or technician may read the image in order to provide a diagnosis.
In PET scanning, for example, a patient is injected with tracer compounds labeled with positron-emitting radionuclides. As the radionuclides decay, they emit a positron which travels through the body a short distance until it collides with an electron. The collision causes both the positron and the electron to be annihilated. In the annihilation process, two gamma rays are emitted.
The gamma rays are emitted at substantially 180° apart from each other. The gamma rays travel out of the body and are detected by the PET scanner. If the two rays reach the scanner within a small time window (e.g. 9 ns) they are considered to be coincident and the spatial locations of each incident ray on the detectors are recorded. However, if one ray is detected without a coincident or with more than one coincident it is discarded.
The line connecting the spatial locations of the coincident rays on the pair of detectors that detected the coincident event is known as a line-of-response (LOR). Once the scan is complete, and all of the LORs are recorded, the number of counts assigned to an LOR joining a pair of crystals is proportional to a line integral of the activity along that LOR. Parallel sets of such line integrals are known as projections. Reconstruction of images from such projections is a problem to which much attention has been paid over the last 30 years, and many analytical and iterative reconstruction schemes exist. Fundamentally, it is desired to ascertain where along the LOR each event occurred, in order to reconstruct an accurate tomographic image.
Image reconstruction in fully 3D medical imaging is routinely performed using either the 3D reprojection method or a rebinning procedure in combination with 2D filtered back projection (FBP). While these algorithms can be realized with relatively low computational cost, the accuracy of the reconstructed images is limited by the approximation implicit in the line integral model on which the reconstruction formulas are based.
In contrast, statistical method can adopt arbitrarily accurate models for the mapping between source volume and sinograms. However, iterative 3D reconstruction represents a daunting computational challenge. One method proposed to overcome this challenge, is using a pre-computed system matrix with axial translation to model span blurring by projection into line-of-response space. While this method saves on computational power, it is still a time consuming task.
It is often necessary to calculate the values of line integrals through a reconstructed image of pixel values. There are many methods, e.g., Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstructive Technique (SIRT), whose purpose is to obtain reconstructions under conditions in which the usual FBP algorithms work poorly.
Back projection is the formal inverse of the projection process. The process originated from work in a number of fields, including radio astronomy and electron microscopy. Simple back projection convolves any f (x, y) with 1/r, thus, the image gets severely blurred and distorted. This is the point-spread function (PSF) for a normal PET. In real systems, the taking of projections has finite resolution characterized by this PSF. To reduce the unwanted blurring effects mentioned above, one could deconvolute with a filter (filtered back projection). In deconvolution, one may cut out unwanted frequencies in Fourier space. The convolution may be equivalently performed in Fourier space where the convolution operation reduces to a simple multiplication. The reconstruction by filtered back projection is a way of filtering the simple back projection by multiplying by v in Fourier space.
In a class of algorithms for calculating projects, know as the Square Pixel Method, the basic assumption is that the object considered truly consists of an array of N×N square pixels, with the image function f(x, y) assumed to be constant over the domain of each pixel. The method proceeds by evaluating the length of intersection of each ray with each pixel, and multiplying the value of the pixel (S).
The major problem of this method is the unrealistic discontinuity of the model. This is especially apparent for rays whose direction is exactly horizontal or vertical, so that relatively large jumps occur in S values as the rays cross pixel-boundaries.
A second class of algorithms for calculating projections is the forward projection method. This method is literally the adjoint of the process of “back projection” of the FBP reconstruction algorithm. The major criticism of this algorithm is that the spatial resolution of the reprojection is lessened by the finite spacing between rays. Furthermore, increasing the number of pixels does not contribute to a reduction in this spacing, but does greatly increase processing time.
A third algorithm for calculating projections, developed by Peter M. Joseph and described in the paper entitled “An Improved Algorithm for Reprojecting Rays Through Pixel Images,” IEEE Transactions On Medical Imaging, Vol. MI-1, No. 3, November 1982 (Joseph's Method), incorporated herein by reference, is similar to the structure of the pixel method. Each given ray K is specified exactly as a straight line. The basic assumption is that the image is a smooth function of x and y sampled on the grid of positions. The line integral desired is related to an integral over either x or y depending on whether the ray's direction lies closer to the x or y axis.
There remains a need in the art for improvement in image reconstruction techniques in order to increase accuracy and resolution and reduce noise.